It is known that spheres have negative type, but only subsets with at most one pair
of antipodal points have strict negative type. These are conditions on the (angular)
distances within any finite subset of points. We show that subsets with at most one
pair of antipodal points have strong negative type, a condition on every probability
distribution of points. This implies that the function of expected distances to points
determines uniquely the probability measure on such a set. It also implies that the
distance covariance test for stochastic independence, introduced by Székely, Rizzo
and Bakirov, is consistent against all alternatives in such sets. Similarly, it allows
tests of goodness of fit, equality of distributions, and hierarchical clustering with
angular distances. We prove this by showing an analogue of the Cramér–Wold
theorem.
Keywords
Cramér–Wold, hemispheres, expected distances, distance
covariance, equality of distributions, goodness of fit,
hierarchical clustering.
